The vanishing of reduced -cohomology for amenable groups can be traced to the work of Cheeger and Gromov in . The subject matter here is reduced -cohomology for , particularly its vanishing. Results for the triviality of are obtained, for example: when and is amenable; when and is Liouville (e.g. of intermediate growth).
This is done by answering a question of Pansu in [34, §1.9] for graphs satisfying certain isoperimetric profile. Namely, the triviality of the reduced -cohomology is equivalent to the absence of non-constant harmonic functions with gradient in ( depends on the profile). In particular, one reduces questions of non-linear analysis (-harmonic functions) to linear ones (harmonic functions with a very restrictive growth condition).
Cite this article
Antoine Gournay, Boundary values, random walks, and -cohomology in degree one. Groups Geom. Dyn. 9 (2015), no. 4, pp. 1153–1184DOI 10.4171/GGD/337